1. INTRODUCTION

Glasses containing metallic nanoparticles (NPs) are extensively studied in order to obtain composite materials with valuable optical properties [1,2,3]. Of particular interest are glasses with silver NPs due to well-defined localized surface plasmon resonance (LSPR) in the visible region of optical extinction spectra [4,5,6,7,8,9,10,11,12]. This phenomenon ensures the localization and preservation of incident light energy on a nanometer scale in space and on a femtosecond scale in time. One of the most significant and rapidly developing directions involves the use of LSPR in Ag NPs to develop laser-active materials having improved luminescent characteristics. In particular, such materials can be used as effective light-sensitizing agents to enhance laser junctions on rare earth (RE) ions introduced into glass [13,14,15,16,17,18]. Based on numerous studies exploring the impact of noble metal NPs on the luminescent characteristics of RE ions, the primary mechanism is shown to involve the enhancement of the local electric field (LEF) due to the presence of LSPR in plasmonic NPs with RE ions located in the field enhancement regions [19,20,21]. This is shown to represent a primary mechanism for noble metal NPs having a size (D ) exceeding 5 nm [20]. In addition, it has been established that the enhancement of LEF through LSPR can also occur for small Ag NPs with D of about 2 nm [22]. For the latter, the field enhancement effect can compete with various mechanisms of energy transfer, both radiative and nonradiative, including energy transfer from multi-phonon relaxation due to the quantum dimensional effect [20] or Foerster resonant energy transfer [23,24,25,26].

The efficiency of sensitization of RE ions by LSPR of plasmonic NPs depends on a number of factors, chiefly:

  1. 1)

    the degree of energy overlap of the LSPR of particles and the absorption energies of RE ions capable of inducing resonance (pumping) [21, 27, 28];

  2. 2)

    the spatial size and localization of regions with enhanced LEF both near individual nanoparticles and their agglomerates;

  3. 3)

    the probability of RE ions being located within such spatial regions. The last two conditions, in turn, depend on the concentration and size of plasmonic NPs, the degree of their agglomeration, and the concentration of RE ions in glass. This concentration, however, should be less or equal to the experimentally determined upper limit for the content of RE ions, preventing the concentration quenching of their luminescence due to the reverse transfer of energy from RE ions to NPs [21, 29,30,31].

This paper presents a theoretical study of the LEF distribution or the number of photonic states N in the vicinity of both single monometallic silver NP and their agglomerates synthesized in a glass matrix [32,33,34,35,36,37,38], depending on the particle size, concentration, and degree of their agglomeration. The dielectric properties of the glass matrix were considered using the refractive index n = 1.5, corresponding to both silicate and zinc phosphate glasses. By using modeling for given concentrations and average sizes of plasmonic NPs in glass, the dependencies of the radial distribution of LEF enhancement sites relative to the surface of the average silver NPs were determined. The optimal synthesis conditions (concentrations of introduced plasmonic metals and RE ions, and glass treatment with temperature or laser irradiation to form NP) proposed on the basis of these dependencies can ensure a high probability of RE ion localization near NP where LEF enhancement is significant.

2. MATERIALS AND CALCULATION METHODS

The areas of LEF enhancement in the vicinity of silver NPs and their agglomerates were calculated using the size and spatial distribution of Ag NPs in silicate glass, which was first subjected to ion exchange followed by irradiation with a UV laser (193 nm) with the number of pulses equal to 5000 [32]. The image of the Ag NP in such a glass is shown in Fig. 1.

Fig. 1.
figure 1

TEM image of a glass sample containing an Ag NP following Ag+ ↔ Na+ ion exchange and subsequent irradiation with 5000 pulses by a UV laser.

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The effect of the LEF induced by LSPR of plasmonic NPs on the fluorescence enhancement of RE ions located near such particles is interpreted using Fermi’s golden rule. This rule defines the probability P of photoabsorption or pumping of RE ions surrounded by a LEF or a field of photons incident on RE ions, whose number N corresponds to the intensity of the LEF:

$${P}_{i\to f}=\frac{2\pi }{\hslash }{\left|\langle {\Psi }_{f}\left|V\right|{\Psi }_{i}\rangle \right|}^{2}\uprho \left({E}_{i\to f}\right).$$
(1)

In this expression, ρ(Ei→f) — density of photonic states around the RE ion with an energy corresponding to the transition of an electron in the RE ion from the initial level of the nucleus i with the wave function Ψi to the final state Ψf; and \(\left|\langle {\Psi }_{f}\left|V\right|{\Psi }_{i}\rangle \right|\) — transition matrix element that occurred due to the influence of the perturbation potential V of exciting photons. Therefore, the final dependence has the form of Pi→f ~ N; ℏ — Planck constant.

The LEF calculations near single monometallic plasmonic nanoparticles and their agglomerates were carried out using two different approaches:

  1. 1)

    the T-matrix method implemented in the T-matrix multiple scattering software code [39] using the developed Python interface to the MSTM code [41, 42];

  2. 2)

    the numerical solution of Maxwell’s equations by the finite difference method using the COMSOL Multiphysics software package [42].

In the second scenario, perfectly matching layers [43] were used to restrict the modeled “box” and suppress the reflection of waves from its boundaries, which are absent in the real material. The T-matrix method was used to identify and eliminate possible instabilities of the numerical solution for Maxwell’s equations by the finite difference method implemented in COMSOL. On the other hand, the finite difference method allows the LEF to be investigated in infinite regular 1D and 2D arrays of plasmon NPs, where the T-matrix method fails to provide an accurate result.

As shown in the TEM image, the LEF calculations for plasmonic Ag nanoparticles distributed in a specific spatial arrangement were carried out for particle agglomerates. These agglomerates represent groups with a finite number of electromagnetically interacting particles, which should be taken into account for the theoretical characterization of the optical absorption spectra of plasmonic Ag NPs distributed in space [41]. The dielectric functions for the bulk silver material were taken from experimental data published elsewhere [44]. As shown in the work [15], the dielectric functions for silver nanoparticles smaller than 10 nm depend on the size. In order to address the impacts of the small size of nanoparticles below 10 nm, it is important to consider corrections to the calculated dielectric function. For nanoparticles with a size of 10 nm, such corrections were not applied in the present study: in this case, their values are insignificant in the visible range for the real part of the dielectric function responsible for photon scattering, while the difference between the imaginary parts of the calculated dielectric functions (responsible for absorption) for such NPs and for bulk silver is minimal at the LSPR wavelength (about 430 nm) [45]. However, the above-mentioned small difference in imaginary parts between the calculated dielectric functions for Ag nanoparticles with a size of 10 nm and for bulk silver at 430 nm has no effect on further findings about the relative variations of the LEF distribution near the considered diverse agglomerates of Ag NPs with a size of about 10 nm.

Figure 2 shows the results of a test calculation comparing the calculations of absorption and scattering contributions calculated using the MSTM and COMSOL software codes to the extinction spectrum of Agglomerate-1, which consists of 11 Ag NPs with the dimensions of D = 10 nm with the distance between their centers of Rij ≥ 15 nm.

Fig. 2.
figure 2

Compared contributions of scattering (b ) and absorption (c) to an extinction spectrum, calculated in COMSOL (solid black curve) and MSTM (dotted red curve) for Agglomerate-1 (a), which consists of 11 silver NPs with dimensions of D = 10 nm, located at a distance of Rij ≥ 15 nm relative to each other.

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The proposed methodology for effectively analyzing the calculated spatial distributions of LEF near plasmonic nanoparticles and their agglomerates can be used to obtain one-dimensional dependencies of the average field enhancement on the distance to the surface of the nearest particle. By enhancing the average LEF falling at a hypothetical location of the RE ion in the immediate vicinity of the plasmonic agglomerate, the methodology facilitates quantitative characterization of the LEF for each of the possible local structural configurations of plasmonic NP systems. Recommendations on preferred nanoparticle type, optimal average size and spacing, and the optimal degree of agglomeration formulated on the basis of this data can be useful for the experimental preparation of state-of-the-art laser materials.

The proposed approach to determining the characteristics of LEF involves the following steps. In the first step, the LEF was calculated for each spatial point near the selected agglomerate of plasmonic NPs where a hypothetical RE ion could be located. The density of such points was selected to ensure monotonic LEF characteristics. In a subsequent step, rather than the conventional representation of the LEF intensity distribution cross plane cross-sections (which depends on their orientation relative to the directions of incidence, polarization of the external field, and their relative orientation to the NP agglomerate), the average LEF enhancement was analyzed across all potential RE ion positions. At that, the RE ions were located at equidistant points Ri from the nearest plasmonic NPs, in the range of Ri ± 0.25 nm, with 0 on the axis of these distances R denoting the NP surfaces within the agglomerate. Fig. 3, schematically depicts the transition from plane section representation to the described depiction of LEF distribution via an averaged N/N0 (R) dependency.

Fig. 3.
figure 3

Schematic representation of LEF distribution near silver NP agglomerate in the form of a “plane section” and N/N0 (R ) using an averaged dependence of LEF enhancement.

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The plasmonic NPs and the associated spatial points used to calculate the LEF were adjusted to accommodate the periodicity of a cubic box. The given number of these points was evenly distributed within the volume of the corresponding regular cubic box. The distance distribution between RE ions and the nearest plasmonic NPs was determined using the Monte Carlo simulation method, which assumes an uncorrelated uniform distribution of hypothetical location points for the RE ions in a glass sample (Fig. 4d). The number of the spatial LEF calculation points and the size of the box were selected in such a way as to ensure the independence of the simulation from their values.

Fig. 4.
figure 4

Studied agglomerates (a and b ), their characteristics (d ), and optical properties (c).

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The number of possible location points of the RE ions as a function of the distance to the surface of the nearest NP was considered using histograms having a bin width of approximately 0.5 nm. The probability density was estimated as the ratio between the number of points in a specific bin at a given distance from the nearest NP, the bin width being divided by the total number of points in the box. The number of plasmonic NPs for LEF calculations was selected based on the concentration of Ag atoms reported in a previous study [32]. The sizes, locations, and degree of nanoparticle agglomeration were selected to replicate the structural configurations and sizes observed in the TEM image of the glass sample shown in Fig. 1 [32]. The formation of Ag nanoparticles in glass involved ion exchange (Ag+ ↔ Na+ ) followed by irradiation with 5000 UV laser pulses (wavelength 193 nm, pulse duration 20 nsec).

3. RESULTS AND DISCUSSION

Figure 1 shows the two main types of silver NPS formed:

  1. 1)

    NPs with a size of D =1–5 nm randomly distributed throughout the sample with an average interparticle distance Rij of approximately 10 nm;

  2. 2)

    NP with a size of D =8– 15 nm located at a distance Rij of approximately 35 nm or closely grouped into the agglomerates with Rij of approximately 15 nm.

In the present study, the different sizes and spatial distribution of Ag NPs represent possible types of deposition of nanoparticles in the glass matrix. This can have various effects on the luminescent properties of embedded RE ions due to the different spatial distribution of LEF near the agglomerates of plasmonic NP.

In order to establish the LEF distributions for the considered groups of silver nanoparticles conforming to the NPs observed in Fig. 1, the calculations in the MSTM program code were carried out using model agglomerates of 10 – 15 NPs as reported in the work [41]. The wavelength λ of the incident light selected close to the wavelength of the LSPR maximum for each type of Ag NP agglomerates was in the range of 425 – 440 nm depending on the considered particle size and their mutual arrangement.

Spatial distributions of LEF near Ag NP agglomerates were calculated for the following groups of particles:

  1. 1)

    Agglomerate-1: particle size of D = 10 nm and distances between particles Rij ≥ 15 nm (values in the lower left part of the TEM image in Fig. 1);

  2. 2)

    Agglomerate-2: D = 10 nm, Rij ≥ 12 nm (lower left corner in Fig. 1);

  3. 3)

    Agglomerate-3: small particles with D =5 nm and Rij ≥ 10 nm (upper right corner in Fig. 1);

  4. 4)

    Agglomerate-4: D = 15 nm, Rij ≥ 35 nm (center of the upper part in Fig. 1), without the small Ag particles located around the large particles;

  5. 5)

    Agglomerate-5: the same large particles as in Agglomerate-4 (D = 15 nm, Rij ≥ 35 nm) surrounded by small Ag NPs with D =5 nm and Rij ≥ 10 nm (the center of the upper part of the TEM image in Fig. 1).

For each of the five agglomerates listed above, each of sections 1 – 4 in Fig. 4 shows the calculations of the distributions N/N0 for LEF (Fig. 4c ) and the probability density w of possible location points of RE ions (Fig. 4d ) as a function of the distance R to the surface of the nearest Ag NP. The calculated LEF distributions and probability densities w of the possible location points of RE ions are averaged in all directions relative to the center of the particles. TEM images of the given groups of Ag NPs are shown in Fig. 4a, and the corresponding parts of the particle agglomerates (with only a portion of each agglomerate being presented) are shown in Fig. 4b. Section 4 (c) of Fig. 4 compares the LEF distributions calculated using Agglomerate-4 (b1, solid black curves) and Agglomerate-5 (b2, dotted blue curves). In section (d ), probability density distributions of possible location points of RE ions corresponding to Agglomerate-4 (blue columns) and Agglomerate-5 (black columns) are compared. Table 1 shows the average enhancement of the LEF depending on the particle size D, the minimum distance between the particle centers Rij, and the component composition of the NP. As shown in Figure 4 and Table 1, the LEF enhancement and the spatial distribution of these points relative to the nearest plasmonic NP are determined by the NP concentration, size, and degree of agglomeration. The lowest LEF enhancement effect, which varies insignificantly with increasing degree of agglomeration, is observed for particles smaller than 5 nm. This is due to the low scattering component present in the optical extinction spectrum of the sample, which determines the characteristics of LEF typical for small NPs. However, LEF increases with an increase in the size and degree of agglomeration for large NPs (≥ 10 nm) (Table 1). Moreover, at an identical concentration of Ag atoms in the particle state, the presence of small NPs (≤ 5 nm) along with relatively large NPs (≥ 10 nm, as in Agglomerate-5) leads to a noticeable decrease in the average LEF enhancement compared with glass having an insignificant number of small NPs (Agglomerate-4).

Table 1. Average LEF Enhancement, Which Falls in One Point of the Hypothetical Location of RE Ion Near Various Agglomerates of Plasmonic Silver NPs at the Wavelength of Exciting Photons Corresponding to the Maximum LSPR for the Considered Agglomerate
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Based on the above considerations, the necessity of using larger NPs in order to achieve the maximum average LEF enhancement can be assumed. However, Fig. 5 shows that the field in the immediate vicinity of the Ag NPs increases with NP size only up to D of approximately 20 nm, followed by a decrease with larger sizes. A similar trend is observed for more distant areas from the NP surface; however, the limiting values of D at which the transition from increase to decrease in the LEF enhancement is observed are larger. Within an agglomerate of such NPs, the probability density of the possible locations of RE ions, being independent of their concentration, exhibits a maximum at a certain distance. Therefore, for each NP agglomerate, an optimal particle size exists that provides the maximum average LEF enhancement.

Fig. 5.
figure 5

N/N0 distribution of LEF enhancement at distances R from the surface of single Ag NP having a size of D = 15 nm (1), D = 20 nm (2), D = 30 nm (3), and D = 50 nm (4).

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As mentioned above, the calculations disregarded the processes of concentration quenching [29] and the energy transfer from the RE ion to the nearest plasmonic NPs, which also leads to quenching of luminescence [47, 48]. The latter can result in a decrease in the LEF enhancement near the NP: the smaller the particle size, the stronger the quenching effect. In the present work, qualitative trends in the spatial distribution of LEF enhancement were identified, depending on the particle size and degree of their agglomeration. In order to verify the above estimate of the optimal size for plasmon particles obtained on the basis of a single-particle model, additional modeling was conducted to suggest the optimal sizes of silver nanoparticles and interparticle spacing within model agglomerates of identical particles.

In order to obtain quantitative structural parameter values for such an optimal configuration of silver nanoparticles, the interaction between the particles was considered. Here the average LEF enhancement per RE ion served as the enhancement criterion, taking into account the spatial volume between the nanoparticles. The modeling results showed that the highest LEF enhancement is characteristic of an Ag NP agglomerate having a size of approximately 25 nm and an average distance between the particle centers of approximately 30 nm.

CONCLUSIONS

Based on the study of the spatial distribution of LEF enhancement near silver plasmonic NPs and their agglomerates, along with the estimates of the average LEF enhancement at potential locations of the RE ion, the main conclusions are as follows:

  • the proposed visual representation of the spatial distribution of the electric field near plasmon NP agglomerates ensures an effective comparative analysis of the field enhancement in hypothetical locations of RE ions;

  • each of the considered local structural configurations of plasmon NP systems is quantitatively characterized by a corresponding average increase in the LEF intensity, which occurs at a hypothetical location of the RE ion in the immediate vicinity of the plasmon NP agglomerate;

  • the enhancement of LEF intensity and the spatial distribution of such enhancement sites relative to the nearest plasmonic NP are determined by the NP concentration, size, and degree of agglomeration, as well as the presence of small (≤ 5 nm) particles in the sample;

  • the optimal configuration of silver nanoparticles in glass for obtaining the maximum average enhancement in the LEF intensity at the hypothetical locations of RE ions is an agglomerate of nanoparticles with sizes slightly exceeding 25 nm and average distances between the centers of particles of approximately 30 nm.

The mechanism presented in this work accounts for the intensity and spatial distribution of LEF near the agglomerates of plasmonic NPs that arises from excitation of LSPR. Such a mechanism is dominant in the enhancement of the luminescence of NPs with a size of ≥ 5 nm. An evaluation of the effects of interactions between the considered NPs and specific RE ions was beyond the scope of this study since the obtained estimates of the average LEF enhancement in the possible locations of the RE ion characterize the optical properties of the plasmon NP agglomerates. Therefore, the optimal concentration of RE ions in the sample remains unstudied. However, the maximum required concentration of RE ions (up to about 1.5 mol.% [29,30,31]), above which the concentration quenching of luminescence commences, can be assumed. Such concentrations of RE ions should lead neither to significant variations in the refractive index or the dielectric function of the glass [29] nor, consequently, to noticeable alterations in the characteristics of the LEF near the NPs and their agglomerates. Since the detailed interactions between RE ions and plasmonic NPs are beyond the scope of this work, additional studies will be required.

The work was supported by a grant from the Russian Science Foundation (RNF Project No. 23-12-00102, https://rscf.ru/project/23-12-00102/).

The authors express their gratitude to P. E. Timoshenko, Cand. Sci. (Phys.-Math.)for assistance and productive discussions.